Back around 45 years ago, as an undergraduate, I took a course in point-set topology based on the first part of Simmons’ book Introduction to Topology and Modern Analysis. The second half of this book was a gentle introduction to the rudiments of functional analysis: Banach and Hilbert spaces, Banach algebras and a glimpse of spectral theory. Since the topology course was so successful and everybody liked Simmons’ text, the professor offered to give a seminar the next semester covering this material.
The topology course was fun, but the follow-up course was a revelation. It dramatically enhanced my understanding and appreciation of linear algebra and also showed me, for what may have been the first time, that very nice things can happen when different branches of mathematics team up. I thought at the time, and still do, that the marriage of linear algebra and analysis that makes up functional analysis was a wonderful way to see this symbiotic relationship.
Despite this, however, functional analysis is rarely taught to undergraduates, and as a result there are not all that many texts on the subject that are accessible to this audience. There are some, however. Simmons’ book is still in print (published by Krieger), and at least two other texts, at a somewhat more sophisticated level, come readily to mind: Saxe’s Beginning Functional Analysis and Rynne and Youngson’s Linear Functional Analysis, both of which are published in Springer series (Undergraduate Texts in Mathematics and Springer Undergraduate Mathematics Series, respectively) that specialize in undergraduate-level books. And now, at an even more sophisticated level, we have the book under review to add to this list.
This text pulls off the neat trick of simultaneously covering quite a lot of functional analysis (the coverage is somewhat broader than in any of the three books mentioned earlier) while still being (for the most part, anyway) accessible to senior-level undergraduates. It begins, as does Simmons’ text, with a look at point-set topology, but here the author does not offer a semester’s worth of material. Instead, he only discusses that which is necessary for future developments. Since all the spaces considered later in the book are normed vector spaces, the author can, and does, limit himself to metric spaces in this (roughly 85-page long) introduction. Topics covered include the basic definitions and properties of metric spaces, convergence and continuity, completeness, separability, compactness and connectedness. Not surprisingly, there is a strong emphasis on function spaces as examples, and, for example, the Arzelà-Ascoli and Stone–Weierstrass theorems are proved, but Tychonoff’s theorem on the product of compact spaces is not.
Part II of the book deals with Banach and Hilbert spaces and their operators. The “big” theorems of the subject (Hahn-Banach, Closed Graph, Uniform Boundedness and Open Mapping) are all proved (in a nice touch, the Hahn-Banach theorem is given both its analytic interpretation in terms of extending functionals and its geometric form in terms of separating hyperplanes). Other standard topics are also discussed: the dual and double dual spaces, the adjoint of an operator, weak and pointwise convergence, and so forth. Compact operators, which Simmons never mentions at all, are also investigated, with integral operators given as examples. The applications of functional analysis are not slighted, and we see, for example, sections on Fourier series and least squares. A final chapter in this part of the book discusses differentiation and integration, and shows how, for example, the notion of a linear operator can be used to define the Fréchet derivative.
The third and final part of the book introduces Banach algebras and uses them to analyze the spectra of operators. The final chapter defines C* algebras, and, among other things, proves the spectral theorems for compact normal operators on a Hilbert space, von Neumann’s spectral theorem for normal operators in terms of spectral measures, and the Gelfand-Naimark theorem. It is with these last two theorems (“often claimed to be the pinnacle of the subject of functional analysis”, according to the text) that the book ends.
One feature that I thought was particularly attractive, especially for undergraduates, was the strong emphasis on specific examples. The dual spaces of many spaces are worked out in detail, for example, and other results (such as the density of polynomials in a number of specific function spaces) are also established. This is valuable both for the student and the instructor, who now has a nice source for many of these well-known-but-sometimes-hard-to-find examples.
Throughout, the author’s writing style is clear, reader-friendly and accessible. (A good background in linear algebra and introductory real analysis should take the reader a long way.) Analogies to finite-dimensional linear algebra are often used to help the student understand the material and put it into context. There are lots of exercises covering a reasonable span of difficulty, and there is a short appendix (about 15 pages long) with hints (but usually not complete solutions) to a number of them. I thought these exercises were quite well-chosen; some pointed out useful things that are not always mentioned explicitly, such as the fact that the Hahn-Banach theorem is essentially trivial for Hilbert spaces.
There is also some attention paid to historical notes; photos and paragraph-long mini-biographies of some of the big names in the subject are scattered throughout the text. Somewhat longer biographies, as in Saxe’s book, would have been nice, but since many books don’t discuss the historical aspects of the subject at all, it seems churlish to complain.
Some parts of the book may prove less accessible to undergraduates than others, of course. One issue that must always be addressed in introductory accounts of functional analysis, for example, concerns the extent to which Lebesgue theory is going to be used. In Simmons, it is not mentioned at all; in Rynne and Johnson, it, according to a review of that book on this site, “keeps sneaking in and being shooed away.” Muscat provides a fairly quick account of the subject in chapter 9 in connection with discussions of various function spaces. The discussion here is sufficiently rapid that someone without prior exposure to this material (such as, for example, most any undergraduate) will likely find it fairly difficult. Likewise, a fair amount of the material at the end of the book, specifically including the spectral theorems, would likely be above the heads of most undergraduates. However, because there is much more in this book than could ever be covered in a single semester, it would seem that this less accessible material could be easily omitted. The book is therefore flexible enough to serve as a text for either senior undergraduates or early graduate students.
It may not, however, be suitable as a text for more sophisticated, upper-graduate level, courses. Designed as an introduction to the subject, it necessarily omits a number of more sophisticated topics (e.g., general topological vector spaces, distributions, unbounded operators) that are covered in more advanced books (e.g., Rudin’s Functional Analysis). Of course, reading this book would provide an excellent introduction to such higher level texts.
All in all, this is a useful and valuable book. I’m glad I own it.
Mark Hunacek (firstname.lastname@example.org) teaches mathematics at Iowa State University.